I'm interested in proving the following proposition ([G], Remark page 48):
Prop: A $\text{Spin}^c$-structure over an oriented vector bundle is equivalent (after stabilizing if the fiber dimension is odd or $\leq 2$) to a complex structure over the $2$–skeleton that can be extended over the $3$–skeleton
The sketch of proof given is the following: First observe that the inclusion $i\colon U(n)\to SO(2n)$ lifts to a map $j\colon U(n)\to \text{Spin}^c(2n)$. For $n \geq 2$, this correspondence is bijective for $2$–complexes and surjective for $3$-complexes, since the map $Bj$ has a $2$–connected fiber. The observation now follows from the fact that restriction induces a bijection from $\text{Spin}^c$–structures to those over the $2$–skeleton extending over the $3$–skeleton. This should conclude the proof, but I'm try to understand why.
I'd like to use the following diagram:
and use the induced map between fibres $F,F'$ together with naturality of the obstruction classes to prove that for a 2-dimensional complex having a Spin^c structure is equivalent of having an almost complex one. The first thing I'm not sure about is what kind of connectivity does the map between fiber have. If that map induces an iso on homotopy groups up to degree $2$ and a surjection up to degree $3$ the first claim should be proved. It's still unclear to me how to conclude from there that "restriction induces a bijection from $\text{Spin}^c$–structures to those over the $2$–skeleton extending over the $3$–skeleton."
Can someone help me shed some light on that?
[G] Robert. E. Gompf $\text{Spin}^c$-structures and homotopy equivalences Geometry and Topology Volume 1 (1997) 41-50. (Here)
